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It has been noticed by many authors that the Schur indices of the irreducible characters of many quasi-simple finite groups are at most 2. A conjecture has emerged that the Schur indices of all irreducible characters of all quasi-simple finite groups are at most 2. We prove that this conjecture cannot be extended to the set of all finite perfect groups. Indeed, we prove that, given any positive integer n, there exist irreducible characters of finite perfect groups of chief length 2 which havedoi:10.1090/s0002-9939-01-06072-5 fatcat:5l5t7d4ilfewrhpfli6p2cl4r4